Theorem lemul2i

Description: Multiplication of both sides of 'less than or equal to' by a positive number. (Contributed by NM, 1-Aug-1999)

	Ref	Expression
Hypotheses	ltplus1.1	⊢ 𝐴 ∈ ℝ
	prodgt0.2	⊢ 𝐵 ∈ ℝ
	ltmul1.3	⊢ 𝐶 ∈ ℝ
Assertion	lemul2i	⊢ ( 0 < 𝐶 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )

Proof

Step	Hyp	Ref	Expression
1		ltplus1.1	⊢ 𝐴 ∈ ℝ
2		prodgt0.2	⊢ 𝐵 ∈ ℝ
3		ltmul1.3	⊢ 𝐶 ∈ ℝ
4		lemul2	⊢ ( ( 𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )
5	1 2 4	mp3an12	⊢ ( ( 𝐶 ∈ ℝ ∧ 0 < 𝐶 ) → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )
6	3 5	mpan	⊢ ( 0 < 𝐶 → ( 𝐴 ≤ 𝐵 ↔ ( 𝐶 · 𝐴 ) ≤ ( 𝐶 · 𝐵 ) ) )